Jomot, that is an interesting observation. Again, I'm trying to be totally neutral. 'Edward' (at the right end of the image) does look a little bit squeezed in, and possibly even with a different pen, (or possibly needing dipping in ink), and Adamson is a little bit higher, but then I noticed that all the surnames are a fraction higher. I've attached a copy for the enjoyment of others. I might be able to visit the actual church later this year.
For what it is worth, I think Tudhoe Jane and Hartlepool Jane are the same, but not all the family share my view, and I agree that it is not conclusive enough for me. YET.
The fact that Tudhoe Jane was raised by Edwin Potter, former blacksmith (probably a colliery blacksmith, not a 'horseshoe' blacksmith) and contractor, married to Anne Adamson, and Hartlepool Jane names her father as an engineer does a lot to swing it for me, but I want more. (I assume that either the registrar assumed her father was called Adamson, rather than Potter, or perhaps Jane said it was Adamson to appear 'more respectable'. She appears as Adamson on the Potter census entries, even though Edwin Potter married Anne Adamson.
Martin